78 
Fishery Bulletin 11 6(1) 
6 
4 
2 
0 - 
0 1 2 3 4 5 6 7 8 9 10 
Age (year) 
Figure 2 
Plot of the true length-at-age (solid line) and estimated length-at- 
age accounting for the observed length-based gear selectivity that 
included an approximation of age-based availability of fish. The es¬ 
timated lengths-at-age are from the example described in Table 1 
(dotted line), from a change in the example reducing the coefficient 
of variation (CV) for the true length-at-age relationship to CV=0.1 
(dotted and dashed line), and from a change in the example where 
the observations are taken from an area with 5% availability of 
age-4+ fish and 100% availability of fish of ages 0-3 (dashed line). 
CJj 
C 
3 
3 4 5 6 
Age (year) 
10 
Length (cm) 
Figure 3 
(A) Plot of true length-at-age (solid line) and the estimated length- 
at-age accounting for the observed length-based selectivity that 
included an approximation of age-based availability. Two different 
true length-based gear selections were used: asymptotic (dotted 
line) and domed shaped (open squares). (B) Plot of the true length- 
based gear selectivity without accounting for age-based availability 
and used in the estimation of length-at-age in plot A. The two ex¬ 
amples depicted are asymptotic (solid line) and domed (dotted line). 
Pi = the observed length-based selec¬ 
tivity as estimated by an inte¬ 
grated model. 
The mean length-at-age for the true popula¬ 
tion is given by 
EiPa.1 ' 
(7) 
Similarly, we can replace p a l with q a t to 
calculate the mean length-at-age for the ob¬ 
served lengths-at-age from the fishery and 
with m a i to calculate the mean length-at-age 
after accounting for the observed selectivity. 
Results 
A bias in the estimate of mean length-at- 
age occurs when expected lengths-at-age ac¬ 
count for the observed selectivity (iq) that 
incorporated an approximation of age-based 
availability in addition to the length-based 
gear selectivity (Fig. 1A). Incorporating the 
approximation results in an alteration of 
the true length-based process (Fig. IB). In 
this example, the asymptotic true length- 
based gear selectivity has the well-known 
effect of observing larger than true fish, and 
this observed bias is unaffected by the age- 
based availability. However after account¬ 
ing for the observed length selectivity that 
includes the additional approximation of 
age-based availability of fish, the selectiv¬ 
ity over-corrects the observed lengths-at-age 
and results in an estimated length distribu¬ 
tion that is shifted to smaller fish (Fig. 1C). 
The magnitude and direction of the ap¬ 
proximation bias on the estimated length- 
at-age is affected by several factors. The 
variability in the length-at-age relationship 
affects the magnitude of the approximation 
bias, and larger variability leads to larger 
bias in the example (Fig. 2). The magnitude 
and direction of the approximation bias is 
also affected by the pattern of age-based 
availability (Fig. 2), and the direction of the 
bias changes if the availability is reversed 
and young fish are fully available and older 
fish are largely unavailable. After the ob¬ 
served selectivity is accounted for, the true 
shape of the length-based gear selectivity 
does not affect the estimated mean length- 
at-age (Fig. 3, A and B). 
Discussion 
In a growing body of research, the effects 
of spatial structure on important model pro¬ 
cesses such as selectivity (Waterhouse et ah, 
2014; O’Boyle, 2016), and the reliability of 
